Question: Subtract the following rational expressions. $\dfrac{2k}{3k-7}-\dfrac{3}{8k}=$
Answer: We can subtract two rational expressions whose denominators are equal by subtracting the numerators and keeping the denominator the same. [Does this fit with how we subtract rational numbers?] When the denominators are not the same, we must manipulate them so that they become the same. In other words, we must find a common denominator. Since the two denominators do not share any common factors, the common denominator is simply the product of these two denominators: $({3k-7})\cdot({8k})$. Let's manipulate the expressions to have that denominator: $\begin{aligned} &\phantom{=}\dfrac{2k}{{3k-7}}-\dfrac{3}{{8k}} \\\\ &=\dfrac{2k\cdot({8k})}{({3k-7})\cdot({8k})}-\dfrac{3\cdot({3k-7})}{({8k})\cdot({3k-7})} \end{aligned}$ [Why did we do that?] Now that both denominators are the same, let's subtract! $\begin{aligned} &\phantom{=}\dfrac{2k\cdot(8k)}{(3k-7)\cdot(8k)}-\dfrac{3\cdot(3k-7)}{(8k)\cdot(3k-7)} \\\\ &=\dfrac{2k\cdot(8k)-3\cdot(3k-7)}{(3k-7)(8k)} \\\\ &=\dfrac{16k^2-9k+21}{(3k-7)(8k)} \end{aligned}$ In conclusion, $\dfrac{2k}{3k-7}-\dfrac{3}{8k}=\dfrac{16k^2-9k+21}{(3k-7)(8k)}$